A spatially periodic structure is a longitudinal array of identical substructures. A method of free and forced vibration analysis of periodic structures with simply supported and guided ends is presented. The computational effort required roughly equals the effort to analyze a single substructure. All natural modes and frequencies ofan extensive structure can be found from a characteristic equation of order equal to the degrees of freedom on one substructure boundary. A modal reduction further simplifies the computations; applications are presented. The method is expected to be useful in structural vibration, sound transmission, and sonic fatigue analysis.